The construction of a non-separable reflexive Banach space on which every operator is the sum of a scalar multiple of the identity operator and an operator of separable range is presented. Using a result of Rao, a sufficient condition is given for Banach spaces with smooth norms to be decomposable. It is shown that operators on Banach spaces of co-dimension one in their biduals are the sum of a scalar multiple of the identity operator and a weakly compact operator. The Banach spaces of bounded operators L(1<sup>1</sup>, 1<sup>p</sup>) (1<p<ꝏ) and L(1<sup>p</sup>, 1<sup>r</sup>), 1 < p ≤ r ≤ p<sup>1</sup> < ꝏ, where 1/p + 1/p<sup>1</sup> = 1, are shown to be primary. The spaces of bounded diagonal operators and compact diagonal operators on a seminormalized Schauder basis β, the multiplier algebras L<sup>d</sub>(X, β) and K<sub>d</sub>(X, β), are introduced and studied. New examples of these multiplier algebras are presented and a theorem of Sersouri is extended. A necessary and sufficient condition is given for c<sub>o</sub> to embed in K<sub>d</sub>(X, β). A sufficient condition is given on a semi-normalized Schauder basis β of a reflexive hereditarily indecomposable Banach space Y to ensure that K<sub>d</sub>(Y, β) has the RNP. It is shown that the algebra L<sub>d</sub>(X, β) is semisimple and that on the algebra K<sub>d</sub>(X, β) derivations are automatically continuous. By representing diagonal operators as stochastic processes a general method of constructing multiplier algebras is given. A non trivial multiplier invariance for the normalized Haar basis of L<sup>1</sup>[0,1] is proved.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:362117 |
| Date | January 1997 |
| Creators | Wark, H. M. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:467c7ec7-d9f1-41cd-9fa9-0f97894ac6a5 |
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